**1. Introduction**

The viscoelastic fluid is a type of non-Newtonian fluid that manifests the viscous and elasticity features under deformation. Sutterby fluid is an example of the viscoelastic fluid, and it well portrays the dilute polymer solutions [1,2]. Specifically, the Sutterby model fluid resembles the shear thinning and shear thickening aspects in high polymer aqueous solutions such as carboxymethyl cellulose (CMC), hydroxyethyl cellulose (HEC) and methyl cellulose (MC) [3]. The dilute polymer solutions have a wide range of functions in industrial practice, for instance, spray applications of agricultural chemicals [4], drag reducers in pipe flows [5], and production of domestic cleaning products [6]. The work of Fujii et al. [7] is one of the earliest studies to address the natural convection boundary layer flow in a Sutterby fluid past a vertical motionless isothermal plane and achieved an excellent comparison with the experimental results. Fujii et al. [8] revisited their work in [7] to investigate the impact of uniform heat flux under the same settings. However, the Sutterby model fluid received less attention from the boundary layer researchers at that time. Later, a new type of heat conductive fluid was introduced by Choi [9] named nanofluid. Nanofluid was also claimed to be a brilliant fluid due to its excellent heat transfer performance in engineering applications such as cooling of electronic appliances, and systems of solar water heating [10]. Nanofluid has now attracted significant interest from researchers, and boundary layer models have been studied under various settings [11–13]. After a long discontinuity in the theoretical works of the Sutterby boundary layer fluid flow, some numerical investigations of the Sutterby fluid under the Cattaneo–Christov heat flux [14,15], Soret and Dufour e ffect [16], peristaltic flow [17–19], squeezed flow [20], Joule heating e ffect [21], homogeneous–heterogeneous reactions [22], and hybrid nanoparticles [23] have been reported recently.

Magnetohydrodynamics (MHD) is another technological conception that is widespread in engineering practice. Electromagnetic casting [24], plasma confinement, and MHD power generation [25] are examples of notable applications. Thermal radiation is a type of energy that works in conjunction with the MHD e ffect. Thermal radiation emits and absorbs energy in the form of waves or molecules through a non-scattering medium. The successful combination of thermal radiation and MHD in an electrically conducting fluid has significant applications in solar power technology and electrical power generation [26]. Acknowledging these applications, researchers began to examine thermal radiation and MHD e ffects in the boundary layer flow past a stretching/shrinking surface, and many theoretical works have been reported. Recently, Sabir et al. [27] explored the stagnation-point flow of a Sutterby fluid with the e ffects of an inclined magnetic field and thermal radiation past a stretching surface, and observed the declining trend of the convective heat transfer with the stronger influence of thermal radiation. Bilal et al. [28] examined the ohmically dissipated Darcy–Forchheimer slip flow of an MHD Sutterby fluid past a radiating stretching sheet and found a decrement in the convective heat transfer with increasing slip e ffects.

By comparison, the boundary layer equations, which were proposed by Prandtl [29], disclosed many invariant closed-form solutions. Prandtl's boundary layer equations can be reduced to a less complicated form that is in a system of ordinary di fferential equations. These boundary layer equations also allow many di fferent types of symmetry groups, of which the Lie group analysis is prominent. Lie group analysis helps to identify the transformation point that represents the given boundary layer equations [30]. In Lie group analysis, the group-invariant solutions are the similarity solutions, and these similarity solutions are used to reduce the independent variables in a fluid flow problem [31]. A special form of the Lie group analysis exists, namely, the scaling group of transformation, and this has been employed by researchers in valuable contributions, for instance, see [32,33].

Regarding studies of stagnation-point flow in a Sutterby fluid, Azhar et al. [34] investigated the effect of entropy generation on the stagnation-point flow of a Sutterby nanofluid past a stretching sheet. Azhar et al. [35] reconsidered the work of [34] by incorporating the Cattaneo–Christov heat flux model and omitting the nanoparticles. Both of the studies of [34,35] solved the flow problem numerically and presented unique solutions. A number of considerable research gaps were found in the theoretical works available in the stagnation-point flow and heat transfer in a Sutterby nanofluid, for instance: inspecting fluid flow behavior and heat transfer characteristics past a shrinking sheet together with the suction e ffect; conducting scaling group analysis; obtaining dual solutions; and performing stability analysis. Thus, the present work is devoted to numerically solving the problem of boundary layer Sutterby nanofluid flow and heat transfer near the stagnation region over a permeable moving (stretching/shrinking) sheet. The fluid flow and heat transfer characteristics under the magnetic and thermal radiation e ffects are observed. Scaling group analysis is employed to obtain the apt similarity transformations so that the complex governing boundary layer equations can be brought to a soluble form. The simplified form of the mathematical model is then solved numerically in the boundary value problem solver or bvp4c function in MATLAB. Two di fferent numerical solutions are identified with the governing parameters' variation. Further, stability analysis is undertaken in the present work to justify the presence of dual solutions. These contributions are essentially original, and all numerical results are presented and discussed in detail.
